Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10018 |
| automorphism | Aut(G) = C2×#72278229 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
235 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,4)(3,9,12,16)(5,13,15,8)(7,14,10,11)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(22,31)(25,32);(1,4,6,2)(3,10,12,7)(5,8,15,13)(9,14,16,11)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10025, #235431, #235433, … |
| maximal normal subgroup of | C2×#10025, #235431, #235433, … |
| maximal quotient of | #103018, #103021, #103023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3013);